Whenever I was little we used to sing:

"Miss Suzie had a steamboat,
the steamboat had a bell, ding ding!
Miss Suzie went to heaven,
the steamboat went to...hell-O operator,
give me number 9, and if you disconnect me,
I'll chop off your be-HIND the fridgerator,
there laid a piece of glass,
Miss Suzie sat upon it and broke her little
ASK me no more questions, tell me no more lies,
the boys are in the bathroom, zipping up their
FLIES are in the meadows, bees are in their hives,
Miss Suzie and her boyfriend are kissing in the
D-A-R-K, D-A-R-K Dark, dark, dark darker than the ocean,
darker than the sea, darker than the underwear my grandma
puts on me. My mother is from Jupiter, my father
is from France, my sister is the stupid one who made me sing this song."

(Not sure how to really spell most of this, so I'm gonna try?)
"Bo, bo see ya-in-taut-in nay, nay, I am BOOM BOOM BOOM,
itty bitty ya-in-taut-in, bo, bo see ya-in-taut-in, bo, bo see ya-in-taut-in BOOM."
---I remember my friends and I always had competitions to see who could sing and do the hand motions the fastest.

"Miss Sue, Miss Sue,
Miss Sue from Alabama, sittin' in a rocker,
Eatin' Betty Crocker, watching the clock go
Tick tock, tick tock Banana rock,
Tick tock, tick tock Banana rock,
A-B-C-D-E-F-G, wash those boys germs off of me,
got a cramp in my side, got a cramp in my side,
don't move!"

"Down down baby, down by the roller coaster,
sweet sweet baby, never never let me go,
shimmy shimmy cocoa puff, shimmy shimmy WOW,
shimmy shimmy cocoa puff, shimmy shimmy BREAKDOWN.
I like coffee, I like tea,
I like the boys and the boys like me,
Grandma grandma, sick in bed,
called the doctor and the doctor said
'Let's get the rhythm of the head DING-DONG!
Let's get the rhythm of the hands (clap twice)
Let's get the rhythm of the feet (stomp twice)
Let's get the rhythm of the HOTTTT DOGGG (move body in circular motion)
Put it all together and what do you get?
DING-DONG, (clap clap) (stomp stomp) HOTTT DOGGG.
Put it all backwards and what do you get?
HOTTT DOGGG, (stomp stomp) (clap clap) DING-DONG!"

"Miss Suzie had a baby, she named him Tiny Tim,
she put him in the bathtub to see if he could swim.
He drank up all the water, he ate up all the soap,
he tried to eat the bathtub, but it wouldn't go down his throat.
Miss Suzie called the doctor, the doctor called the nurse,
the nurse called the lady with the alligator purse (says matter-of-factly everytime you say this part)
In came the doctor,
in came the nurse,
in came the lady with the alligator purse.
"Flu" said the doctor,
"Allergies" said the nurse,
nothing said the lady with the alligator purse.
Out went with the doctor,
out went with the nurse,
BOOM went with the lady with the alligator purse!"

"Miss Mary Mack, Mack, Mack!
All dressed in black, black, black!
With silver buttons, buttons, buttons
All down her back, back, back
She asked her mother, mother, mother
For fifty cents, cents, cents
To see the elephants, elephants, elephants
Jump over the fence, fence, fence
They jumped so high, high, high
They touched the sky, sky, sky
And didn't come back, back, back
Till the fourth of July, July, July."

"Double double this, this,
double double that, that.
Double this, double that,
double double this that!"
(sometimes we'd substitute with words such as
Double double straw, straw,
double double berry, berry.
double straw, double berry,
double double strawberry!
Make hands into fists, hit fists with the other person as the two say "double double" then open up hands so fingers are pointing up, clap hands with other person-double high five basically- at word "this" and then palms facing towards you at "that"-so backs of hands are hitting the backs of the other person's hands.)

"Down by the banks of the hanky panky,
where the bull-frogs jump from bank to banky,
with an Ooh, eee, ooh ahhh, he's off his lily with a big
KER-PLOP!"
(this was always fun to do in Girl Scouts, we'd sit in one big circle, hand in hand basically, slap each other's hands around the circle...whoever's hand was slapped at PLOP was out of the circle.)

"Apples on a stick, make me sick,
makes my heart go 246.
Not because I'm pretty,
not because I'm clean,
not because I kiss the boy behind the magazine.
Hey girl, wanna pick a fight?
Meet me on the corner on a Saturday night.
She can wibble, she can wobble,
she can do the splits, but I betcha $5 she can't do this.
Close your eyes, and count to 10,
don't say 8 or you have to kiss your boyfriend!"
(We were always teasing each other if one of us said 8, and saying 'Oh you have to go kiss the boy you like now!!' TORTURE for a 7 year old!)

"I don't wanna go to Mexico no more, more, more,
there's a big fat policeman at my door, door, door.
He took me by the hips, kissed me on the lips,
I don't wanna go to Mexico no more, more, more.

I WANNA go to Mexico some more, more, more,
there's a real cute guy at the door, door, door.
He grabbed me by the hips, kissed me on the lips,
I WANNA go to Mexico some more, more, more!"

Or:

A bunch of us standing in a circle, one would go in the middle, and skip around the circle, and we'd sing:
"Little Sally Walker, walking down the street,
she didn't know what to do, so she stopped in front of me
and said 'Hey girl (or boy) do your thang, do your thang,
SWITCH. Hey girl/boy do your thang, do your thang SWITCH."
(The kid in the middle had to stop in front of someone at the line "Stopped in front of me" and do some crazy dance move or whatever, then the kid he/she stopped in front of me jumped in the middle of the circle at the first "SWITCH" and had to copy the other. Then he/she would be the next person skipping around the circle.)

My friends and I liked to sing these while we jumped rope:
"Cinderella, dressed in yella,
Went upstairs to kiss a fella,
Made a mistake, kissed a snake,
How many doctors will it take?
1-2-3-4...."
(proceed to count until your foot hits the rope or the two others twirling the rope purposefully swings it so you can't jump it.)

"Teddy Bear, Teddy Bear,
Turn around (turn around)
Teddy Bear, Teddy Bear,
Touch the ground (touch the ground)
Teddy Bear, Teddy Bear,
Tie your shoe (hit your shoe)
Teddy Bear, Teddy Bear,
How old are you?
1-2-3-4..........."